3 Description

F07MPF (ZHESVX) performs the following steps:

If FACT='N', the diagonal pivoting method is used to factor A. The form of the factorization is A=UDUH if UPLO='U' or A=LDLH if UPLO='L', where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is Hermitian and block diagonal with 1 by 1 and 2 by 2 diagonal blocks.

If some dii=0, so that D is exactly singular, then the routine returns with INFO=i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, INFO=N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below.

The system of equations is solved for X using the factored form of A.

Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.

On entry: the first dimension of the array X as declared in the (sub)program from which F07MPF (ZHESVX) is called.

Constraint:
LDX≥max1,N.

14: RCOND – REAL (KIND=nag_wp)Output

On exit: the estimate of the reciprocal condition number of the matrix A. If RCOND=0.0, the matrix may be exactly singular. This condition is indicated by INFO>0 and INFO≤N. Otherwise, if RCOND is less than the machine precision, the matrix is singular to working precision. This condition is indicated by INFO=N+1.

On exit: if INFO=0 or N+1, an estimate of the forward error bound for each computed solution vector, such that x^j-xj∞/xj∞≤FERRj where x^j is the jth column of the computed solution returned in the array X and xj is the corresponding column of the exact solution X. The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error.

On exit: if INFO=0 or N+1, an estimate of the component-wise relative backward error of each computed solution vector x^j (i.e., the smallest relative change in any element of A or B that makes x^j an exact solution).

On entry: the dimension of the array WORK as declared in the (sub)program from which F07MPF (ZHESVX) is called.

LWORK≥max1,2×N, and for best performance, when FACT='N', LWORK≥max1,2×N,N×nb, where nb is the optimal block size for F07MRF (ZHETRF).

If LWORK=-1, a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued.

6 Error Indicators and Warnings

If INFO=-i, the ith argument had an illegal value. An explanatory message is output, and execution of the program is terminated.

INFO>0 and INFO≤N

If INFO≤N, di,i is exactly zero. The factorization has been completed, but the factor D is exactly singular, so the solution and error bounds could not be computed. RCOND=0.0 is returned.

INFO=N+1

D is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of RCOND would suggest.

7 Accuracy

For each right-hand side vector b, the computed solution x^ is the exact solution of a perturbed system of equations A+Ex^=b, where

E1=OεA1,

where ε is the machine precision. See Chapter 11 of Higham (2002) for further details.

If x^ is the true solution, then the computed solution x satisfies a forward error bound of the form

x-x^∞x^∞≤wccondA,x^,b

where
condA,x^,b=A-1Ax^+b∞/x^∞≤condA=A-1A∞≤κ∞A.
If x^ is the jth column of X, then wc is returned in BERRj and a bound on x-x^∞/x^∞ is returned in FERRj. See Section 4.4 of Anderson et al. (1999) for further details.

8 Further Comments

The factorization of A requires approximately 43n3 floating point operations.

For each right-hand side, computation of the backward error involves a minimum of 16⁢n2 floating point operations. Each step of iterative refinement involves an additional 24⁢n2 operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required. Estimating the forward error involves solving a number of systems of equations of the form Ax=b; the number is usually 4 or 5 and never more than 11. Each solution involves approximately 8⁢n2 operations.